P -adic subsets whose factorials satisfy a generalized Legendre formula

Abstract

Amice studied the notion of a regular compact subset in a local field K, with valuation v and maximal ideal 𝔐. In her work, she introduced the notion of well distributed sequences and showed that every regular compact subset S admits well distributed sequences and that its factorial sequence (n!S) satisfies a generalized Legendre formula: ν ( n ! s ) = ∑ i = 1 i = ∞ [ n q i ] for every integer n and where qi denotes the number of classes of S modulo 𝔐i. In this article, in more general settings, we show the converse assertions. More precisely, we prove that, for every precompact subset of any discrete valuation domain V, the following assertions are equivalent: the topological closure of S is a regular subset, S admits a very well distributed sequence, S satisfies the generalized Legendre formula.